Spaces of metrics with positive spectral scalar curvature

[2607.28467] Spaces of metrics with positive spectral scalar curvature Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Differential Geometry arXiv:2607.28467 (math) [Submitted on 30 Jul 2026 ( v1 ), last revised 14 Aug 2026 (this version, v2)] Title: Spaces of metrics with positive spectral scalar curvature Authors: Gioacchino Antonelli , Georg Frenck , Bernhard Hanke View a PDF of the paper titled Spaces of metrics with positive spectral scalar curvature, by Gioacchino Antonelli and Georg Frenck and Bernhard Hanke View PDF HTML (experimental) Abstract: Let $n\geq2$ and let $M^n$ be a closed connected smooth manifold. Let $R^\gamma(M)$ be the space of smooth Riemannian metrics $g$ on $M$ for which the generalized conformal Laplace operator $-\gamma\Delta_g+\mathrm{R}_g$ is strictly positive. We prove that if $n=2$ and $\gamma>0$, or if $n\ge3$ and $0< \gamma \leq 4(n-1)/(n-2)$, the inclusion $R^0(M)\hookrightarrow R^\gamma(M)$ is a homotopy equivalence, thus generalizing, to all dimensions and in the maximal range, the results of Botvinnik--Rosenberg and Li--Mantoulidis. Then, we prove that if $n\ge3$ and $\gamma>4(n-1)/(n-2)$, the space $R^\gamma(M)$ is contractible, and hence nonempty. This solves a homotopy-theoretic strengthening of a conjecture of Gromov (Conjecture 3, Section 6.1.2, "Four Lectures on Scalar Curvature") in the maximal possible coefficient range. Concerning Gromov's conjecture we also treat the equivariant case and the case of manifolds with boundary. Comments: 21 pages. Comments are welcome! v2: Title changed and minor edits Subjects: Differential Geometry (math.DG) Cite as: arXiv:2607.28467 [math.DG] (or arXiv:2607.28467v2 [math.DG] for this version) https://doi.org/10.48550/arXiv.2607.28467 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Gioacchino Antonelli [ view email ] [v1] Thu, 30 Jul 2026 16:25:10 UTC (26 KB) [v2] Fri, 14 Aug 2026 20:58:23 UTC (26 KB) Full-text links: Access Paper: View a PDF of the paper titled Spaces of metrics with positive spectral scalar curvature, by Gioacchino Antonelli and Georg Frenck and Bernhard Hanke View PDF HTML (experimental) TeX Source view license Current browse context: math.DG < prev | next > new | recent | 2026-07 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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